Scaling of quantum capacitances and valence electron detachment energies is studied for icosahedral and nonicosahedral fullerenes. Scaling trends are considered from zero to infinite average radius, where a fullerene's local surface properties are similar to those of graphene. Detailed density functional theory calculations are performed to determine the geometries and detachment energies of icosahedral fullerenes, while values of these quantities are obtained for nonicosahedral species from previously published experimental results. Strongly linear, quasiclassical scaling versus average radii rn is seen for the quantum capacitances, but on two di↵erent scaling lines for icosahedral and nonicosahedral species, respectively. By contrast, nonclassical, nonlinear scaling versus 1/rn is seen for the electron detachment energies-i.e., the valence ionization potentials and electron a nities. This nonlinearity is not accounted for by classical theories that are used to explain trends in electronic properties of fullerenes and usually give accurate quantitative estimates. Instead, simple quantum equations are derived to account for nonlinearities in the metal-particle-like electron detachment energy scaling and to show that these are responsible for nonclassical, nonzero intercepts in the capacitance scaling lines of the fullerenes. Last, it is found that points representing the carbon atom and the graphene limit lie on scaling lines for icosahedral fullerenes, so their quantum capacitances and their detachment energies scale smoothly from one C atom, to C 60, to graphene.

Scaling of quantum capacitances is explored for lanthanide and actinide atoms. For lighter atoms, quantum capacitances have been seen to scale linearly with mean radii of the atoms' outermost occupied orbitals. This scaling law is used to analyze two recent, differing sets of theoretical calculations for lanthanide electron affinities A. Consistent with the scaling law, A values predicted by O'Malley and Beck for lanthanides [Phys. Rev. A 78, 012510 (2008)], using a relativistic configuration interaction (RCI) method, produce capacitances that scale with the atoms' mean radii along a single line to a high degree of confidence. Similar linear scaling behavior also results for the actinides from O'Malley and Beck's RCI calculations of their A values. However, lanthanide A values predicted by Felfli et al. [Phys. Rev. A 81, 042707 (2010)], using a Regge-pole approach, unexpectedly produce capacitance scaling along two different lines for atoms with similar neutral electron configurations. Both types of linear capacitance scaling are internally consistent, though, and do not serve to determine definitively which set of electron affinity predictions for the lanthanides is likely to be more accurate. Still, evidence from this and prior capacitance scaling investigations tends to favor the O'Malley and Beck results. In addition, linear capacitance scaling for the actinides is applied to estimate the previously unknown A values for Fm and Md as 0.007 eV and −0.006 eV, respectively.